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Isaac Newton Institute for Mathematical Sciences

Plane Wave Discontinuous Galerkin Methods

Authors: Ralf Hiptmair (ETH Zuerich), Ilaria Perugia (University of Pavia)

Abstract

Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation $-\Delta u - \omega^{2}u=f$ are afflicted with the so-called pollution phenomenon: though for sufficiently small $h\omega$ an accurate approximation of $u$ is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wave functions $\boldsymbol{x}\mapsto \exp(i\omega\boldsymbol{d}\cdot\boldsymbol{x})$, $|\boldsymbol{d}|=1$, into the trial spaces. Prominent examples of such methods are the plane wave partition of unity finite element method of Babuska and Melenk, and the ultra-weak Galerkin discretization due to Cessenat and Despres. Both perform well in computations.

It turns out that the latter method can be recast as a special so-called discontinuous Galerkin (DG) method employing local trial spaces spanned by a few plane waves. This perspective paves the way for marrying plane wave approximation with many of the various DG methods developed for 2nd-order elliptic boundary value problems. We have pursued this for a generic mixed DG method and a primal DG method which generalized the ultra-weak scheme.

For these methods we have developed a convergence analysis for the $h$-version, which achieves convergence through mesh refinement. Key elements are approximation estimates for plane waves and sophisticated duality techniques. The latter entail estimating how well local plane waves can approximate the solution of a dual problem. Unfortunately, we could not help invoking general polynomial estimates in Sobolev spaces for this purpose. This incurs unsatisfactory pollution-affected final error bounds $O(\omega^{2}h)$. On the other hand, a more detailed 1D analysis confirms that the plane wave DG method does not suffer from pollution. To bridge the gap is an open theoretical challenge.